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On Different Geometric Formulations of Lagrangian Formalism Published in Differential Geom. Appl., 10 n. 3 (1999), 225–255 Version with some corrections/improvements On different geometric formulations of Lagrangian formalism Raffaele Vitolo1 Dept. of Mathematics “E. De Giorgi”, Universit`adi Lecce, via per Arnesano, 73100 Italy email: [email protected] Abstract We consider two geometric formulations of Lagrangian formalism on fibred manifolds: Krupka’s theory of finite order variational sequences, and Vinogradov’s infinite order variational sequence associated with the C–spectral sequence. On one hand, we show that the direct limit of Krupka’s variational bicomplex is a new infinite order variational bicomplex which yields a new infinite order variational sequence. On the other hand, by means of Vinogradov’s C–spectral sequence, we provide a new finite order variational sequence whose direct limit turns out to be the Vinogradov’s infinite order variational sequence. Finally, we provide an equivalence of the two finite order and infinite order variational sequences up to the space of Euler–Lagrange morphisms. Key words: Fibred manifold, jet space, infinite order jet space, variational bicomplex, variational sequence, spectral sequence. 1991 MSC: 58A12, 58A20, 58E30, 58G05. 1 This paper has been partially supported by Fondazione F. Severi, INdAM ‘F. Severi’ through a senior research fellowship, GNFM of CNR, MURST, University of Florence. 1 2 On formulations of Lagrangian formalism Introduction The theory of variational bicomplexes can be regarded as the natural geometrical set- ting for the calculus of variations [1, 2, 10, 11, 15, 19, 20, 21, 22, 23, 24]. The geometric objects which appear in the calculus of variations find a place on the vertices of a varia- tional bicomplex, and are related by the morphisms of the bicomplex. Such morphisms are closely related to the differential of forms. Moreover, the global inverse problem is solved in this context. The purpose of this paper is to compare Krupka’s finite order formulation [10] to the infinite order formulation by Vinogradov [23, 24]. Krupka’s finite order variational sequence is produced when one quotients the de Rham sequence on a finite order jet space by means of an intrinsically defined sub- sequence. So, the morphisms of this bicomplex are either the differential of forms, or inclusions, or quotient morphisms. A finite order formulation of variational bicomplexes can help in keeping trace of the order of the geometric objects involved at each vertex of the bicomplex. But this yields several technical difficulties. For an intrinsic analysis of this theory, based on the structure form on jets [13] and the first variation formula [9], see [28]. The formulation of Vinogradov is carried on by means of the C–spectral sequence. This is a very general framework, by which one can formulate the variational sequences also in the case of the spaces of infinite jets of m dimensional submanifolds of a given m + n dimensional manifold. Moreover, C–spectral sequences play an important role in the theory of ordinary and partial differential equations, and in quantum mechanics and field theory. Here, we consider the variational sequence associated with the C–spectral sequence of the infinite order de Rham exact sequence. Roughly speaking, the infinite order de Rham exact sequence is made by forms on jet spaces of any order. This is a wiewpoint that allows to skip several hard technical difficulties. In fact, one has not to worry about the order of the objects or the operators. The relationship between Tulczyjew’s and Vinogradov’s formulations has been analysed in [4]. Here, we give a new finite order formulation of variational sequences using the C–spectral sequence on finite order jet spaces. The direct limit of this finite order C–spectral sequence turns out to be Vinogradov’s infinite order C–spectral sequence. Then, we evaluate the direct limit of Krupka’s variational bicomplex, finding a new infinite order variational sequence. Finally, we do a comparison of both finite and infinite order variational sequences finding that they are isomorphic up to the space of Euler–Lagrange morphisms. So, the logical scheme of this paper is summarised by the following diagram New infinite–order ≈ Vinogradov’s infinite–order formulation - formulation 6 6 lim→ lim→ Krupka’s finite–order ≈ New finite–order formulation - formulation R. Vitolo 3 A final discussion has to be devoted to the language used in the paper. While Krupka’s approach is carried on by means of the language of sheaves (see, for example, [29]), Vinogradov’s approach uses an algebraic language (see [8] and references therein). Here, in order to compare the above approaches with a unique language, we found easier to left Krupka’s approach unchanged and to use a presheaf approach for the C–spectral sequence. In fact, the C–spectral sequence is defined in the category of differential groups (Appendix B), but, in our case, it can be carried on easily to presheaves. Of course, the algebraic language is a very powerful and natural tool, and in the next future much more could be said about finite order C–spectral sequences, for example in the case of jets of submanifolds. We observe that a basic introduction to spectral sequences is provided in Appendix B, in order to make the paper self–contained. We end the introduction with some mathematical conventions. In this paper, mani- folds are connected and C∞, and maps between manifolds are C∞. Morphisms of fibred manifolds (and hence bundles) are morphisms over the identity of the base manifold, unless otherwise specified. We make use of definitions and results on presheaves and sheaves from [29]. In par- ticular, we are concerned only with (pre)sheaves of IR–vector spaces, hence ‘(pre)sheaf morphism’ stands for morphism of (pre)sheaves of IR–vector spaces. We denote by SU the set of sections of a (pre)sheaf S over a topological space X defined on the open subset U ⊂ X. We recall that a sequence of (pre)sheaves over X is said to be exact if it is locally exact (see [29] for a more precise definition). If A, B are two sub(pre)sheaves of a sheaf S, then the wedge product A∧B is defined to be the sub(pre)sheaf of sections 2 of ∧S generated by wedge products of sections of A and B. We recall that a sheaf S over X is said to be soft if each section defined on a closed subset C ⊂ X can be extended to a section defined on any open subset U such that C ⊂ U. Moreover, S is said to be fine if it admits a partition of unity. A fine sheaf is also a soft sheaf. The sheaf of sections of a vector bundle is a fine sheaf, hence a soft sheaf. m Let {Sn}n∈N be a family of (pre)sheaves and {ιn : Sn → Sm}n,m∈N,n≤m be a family of injective (pre)sheaf morphisms such that, for all n,m,p ∈ N, n ≤ m ≤ p, we have p m p n ιm ◦ ιn = ιn and ιn = idSn . We say {Sn} to be an injective system. We define the direct limit of the injective system to be the (pre)sheaf S := Sn ∼ , nG∈N ′ where ∼ is the equivalence relation defined as follows. For each s ∈ Sn and s ∈ Sn′ , if ′ ′ n′ ′ n ≤ n , then s ∼ s if and only if ιn (s)= s . Acknowledgements. I would like to thank I. Kol`aˇr, D. Krupka, M. Modugno, J. Stefanekˇ and A. Vinogradov for stimulating discussions. Diagrams have been drawn by P. Taylor’s diagrams macro package. 4 On formulations of Lagrangian formalism 1 Jet spaces In this section we recall some facts on jet spaces. We start with the definition of jet space, then we introduce the contact maps. We study the natural sheaves of forms on jet spaces which arise from the fibring and the contact maps. Finally, we introduce the horizontal and vertical differential of forms on jet spaces. Jet spaces Our framework is a fibred manifold π : Y → X , with dim X = n and dim Y = n + m. We deal with the tangent bundle T Y → Y , the tangent prolongation Tπ : T Y → T X and the vertical bundle V Y := ker Tπ → Y . Moreover, for 0 ≤ r, we are concerned with the r–th jet space JrY ; in particular, we set J0Y ≡ Y . We recall the natural fibrings r Y Y r Y X πs : Jr → Js , π : Jr → , and the affine bundle r Y Y πr−1 : Jr → Jr−1 associated with the vector bundle r ∗ ⊙ T X ⊗ V Y → Jr−1Y , Jr−1Y for 0 ≤ s ≤ r. A detailed account of the theory of jets can be found in [13, 11, 17]. Charts on Y adapted to the fibring are denoted by (xλ, yi). Greek indices λ,µ,... run from 1 to n and label base coordinates, Latin indices i,j,... run from 1 to m and λ i label fibre coordinates, unles otherwise specified. We denote by (∂λ,∂i) and (d ,d ), respectively, the local bases of vector fields and 1–forms on Y induced by an adapted chart. We denote multi–indices of dimension n by underlined latin letters such as p = (p1,...,pn), with 0 ≤ p1,...,pn; by identifying the index λ with a multi–index according to λ ≃ (p1,...,pi,...,pn) ≡ (0,..., 1,..., 0) , we can write p + λ =(p1,...,pi + 1,...,pn) . We also set |p| := p1 + ··· + pn and p! := p1! ...pn!. Y 0 i The charts induced on Jr are denoted by (x , yp), with 0 ≤ |p| ≤ r; in particular, i i Y if |p| = 0, then we set y0 ≡ y .
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